Bounding the Maximal Height of a Diffusion by the Time Elapsed

Let X=(X t ) t≥0 be a one-dimensional time-homogeneous diffusion process associated with the infinitesimal generator $$\mathbb{L}_X = \mu (x)\frac{\partial }{{\partial x}} + \frac{{\sigma ^2 (x)}}{2}\frac{{\partial ^2 }}{{\partial x^2 }}$$ where x↦μ(x) and x↦σ(x)>0 are continuous. We show how the question of finding a function x↦H(x) such that $$c_1 E(H(\tau )) \leqslant E(\mathop {\max }\limits_{0 \leqslant {\text{t}} \leqslant \tau } \left| {X_t } \right|) \leqslant c_2 E(H(\tau ))$$ holds for all stopping times τ of X relates to solutions of the equation: $$\mathbb{L}_X (F) = 1$$ Explicit expressions for H are derived in terms of μ and σ. The method of proof relies upon a domination principle established by Lenglart and Itô calculus.